莫尔斯理论 英文PDF电子书下载

PART Ⅰ．NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD 1

1．Introduction 1

2．Definitions and Lemmas 4

3．Homotopy Type in Terms of Critical Values 12

4．Examples 25

5．The Morse Inequalities 28

6．Manifolds in Euclidean Space:The Existence of Non-degenerate Functions 32

7．The Lefschetz Theorem on Hyperplane Sections 39

PART Ⅱ．A RAPID CORSE IN RIEMANNIAN GEOMETRY 43

8．Covariant Differentiation 43

9．The Curvature Tensor 51

10．Geodesics and Completeness 55

PART Ⅲ．THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS 67

11．The Path Space of a Smooth Manifold 67

12．The Energy of a Path 70

13．The Hessian of the Energy Function at a Critical Path 74

14．Jacobi Fields:The Null-space of E＊＊ 77

15．The Index Theorem 83

16．A Finite Dimensional Approximation to Ωc 88

17．The Topology of the Full Path Space 93

18．Existence of Non-conjugate Points 98

19．Some Relations Between Topology and Curvature 100

PART Ⅳ．APPLICATIONS TO LIE GROUPS AND SYMMETRIC SPACES 109

20．Symmetric Spaces 109

21．Lie Groups as Symmetric Spaces 112

22．Whole Manifolds of Minimal Geodesics 118

23．The Bott Periodicity Theorem for the Unitayy Group 124

24．The Periodicity Theorem for the Orthogonal Group 133

APPENDIX．THE HOMOTOPY TYPE OF A MONOTONE UNION 149